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Slip and chemical reaction effects on the peristaltic rheology of a viscoplastic liquid in different wave frames: application of a high-permeability medium

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Abstract

Nanofluid flow due to artificial phenomena such as sinusoidal contraction-relaxation of flexible walls has a significant application in various bio-medical industries, chemical engineering, and new emerging technologies. The present analysis explores the influences of slip conditions and chemical reactions on the viscoplastic nanofluid rheology due to peristaltic pumping through a non-uniform (divergent) channel adjacent to a saturated, non-Darcian, high-permeability porous medium. The Casson fluid is used as a viscoplastic liquid in the current investigation. Mathematical formulations are performed by utilizing Cartesian coordinates. This study is based on the greater wavelength hypothesis and the creeping principle. High permeability is considered to symbolize topographically small-sized porous media. Two external forces are present in the momentum equation: one is the Darcian force, and the second is the Forchheimer inertial force. Scaling variables are used to make the rheological system dimensionless. The non-dimensional flow system is expressed in the PDEs and transformed into the ODEs by using biological approximations. The impacts of chemical reaction parameter \(\left(\chi \right)\), Darcy number \(({\text{DA}})\), second (higher) order velocity slip parameters \((b1,b2)\), thermal slip parameter \((b3)\), concentration slip parameter \((b4)\), Forchheimer number \(({\text{Fs}})\), Casson parameter \((\Gamma )\), Brownian motion \(({\text{nb}})\), thermophoresis \(({\text{nt}})\), heat source/sink parameter \((\sigma )\), Radiation parameter \(({\text{Rn}})\), Brinkman number \(({\text{br}})\), Grashof number \(({\text{GR}})\), local nanoparticle Grashof number \(({\text{GM}})\), and Prandtl number \(({\text{pr}})\) on the velocity profile \(\left(u\left(\omega \right)\right)\), stream function \(\left(\psi \left(\omega \right)\right)\), temperature profile \(\left(\phi \left(\omega \right)\right)\), and nanoparticles mass concentration phenomena \(\left(\Theta \left(\omega \right)\right)\) are studied. Four unique sorts of peristaltic waves (Cosine, Sawtooth, Sine, and Square waves) are utilized to break down the flow. The higher intensity of the Forchheimer force strongly disturbed the parabolic shape of the axial velocity. The thermal enhancement is founded on increasing Brownian motion and thermophoresis parameters. Asymmetric nature is found in graphs due to the non-uniform nature of the flow channel. The contrast among viscous liquid and Casson fluid is also discussed. The key advantage of this investigation is that it is productive in differentiating the role of various natures of peristaltic waves in biological fluids’ motion. This model is productive for the thermal enhancement of mechanical and chemical rheological processes.

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Abbreviations

a :

Half-with of the non-uniform (divergent) channel

\(\Omega\) :

Wave speed

\({\mu }_{0}\) :

Plastic dynamic viscosity of the non-viscous fluid

\({\in }_{nm}\) :

\((m,n)-{\text{th}}\) element of deformation rate

\({\pi }_{0}\) :

Critical point of \(\pi\)

\({\Theta }_{1}\) :

Concentration at upper wall

\({\Theta }_{0}\) :

Concentration at lower wall

\(\widetilde{U,}\widetilde{V}\) :

Velocity components in axial and transverse directions

\(\widetilde{S}\) :

Cauchy stress tensor

\(\tau\) :

Ratio effective heat capacity of nanoparticles to effective heat capacity of fluid

\(\mu\) :

Dynamic viscosity of the fluid

\(G\varepsilon\) :

Thermal expansion coefficient

\({\widetilde{S}}_{\widetilde{X}\widetilde{X}},{ \widetilde{S}}_{\widetilde{X}\widetilde{\omega } }\& {\widetilde{S}}_{\widetilde{\omega }\widetilde{\omega }}\) :

Components of the extra stress tensor

\({q}_{0}\) :

Heat source/sink parameter

\(\Delta\) :

Mean absorption coefficient

\({D}_{{\text{B}}}\) :

Brownian diffusion coefficient

\(\varsigma\) :

Stefan–Boltzmann constant

\(\widetilde{M}\) :

Dimensional non-uniform parameter

\(\widetilde{Z}\) and \(-\widetilde{Z}\) :

Upper and lower walls of the non-uniform channel

\({P}_{0}\) :

Yield stress (fluid or material property corresponding to yield point) of the liquid

\(\pi\) :

Product of deformation rate with itself

\({\phi }_{1}\) :

Temperature at the upper wall

\({\phi }_{0}\) :

Temperature at the lower wall

\(\widetilde{X},\) \(\widetilde{\omega }\) :

Flow coordinates in axial and transverse directions,

\(\widetilde{P}\) :

Pressure

\({\varphi }_{{\text{f}}}\) :

Fluid density

\({(\varphi C)}_{{\text{f}}}\) :

Effective heat capacity of fluid

\({k}^{{\text{p}}}\) :

Permeability of the porous medium

\({K}_{1}\) :

Chemical reaction parameter

\(\gamma\) :

Thermal diffusivity

\({C}_{{\text{f}}}\) :

Heat capacity of fluid

\({(\varphi C)}_{{\text{p}}}\) :

Effective heat capacity of nanoparticles

\({D}_{{\text{T}}}\) :

Thermophoretic diffusion coefficient

\(G\varepsilon \mathrm{^{\prime}}\) :

Concentration expansion coefficient

\(x\) :

Axial coordinate of flow geometry

\(t\) :

Time

v :

Velocity component in the transverse direction

\(\phi\) :

Temperature profile

\({\text{nt}}\) :

Thermophoresis parameter

\({\text{GM}}\) :

Grashof number

\({\text{Re}}\) :

Reynold’s number

\({\text{DA}}\) :

Darcy number

\({\text{pr}}\) :

Prandtl number

\(A\) :

Amplitude ratio

\({\text{Fs}}\) :

Forchheimer drag

\(m\) :

Non-uniform (divergent) parameter

\(\chi\) :

Chemical reaction parameter

\({\text{Ec}}\) :

Eckert number

\(\omega\) :

Transverse coordinate of flow geometry

\(u\) is the velocity component in the axial direction:

\(u\) is the velocity component in the axial direction

\(p\) :

Pressure

\(\Theta\) :

Mass concentration

\({\text{nb}}\) :

Brownian motion parameter

\({\text{GR}}\) :

Local temperature Grashof number

\(\delta\) :

Wavelength

\({\text{Rn}}\) :

Thermal radiation parameter

\(\sigma\) :

Dimensionless heat source/sink parameter

\({b}_{j} (j=1-4)\) :

Slip parameters (1 and 2 stand for velocity slip, 3 stand for thermal, and 4 stand for concentration)

\({\text{br}}\) :

Brinkman number

\({s}_{ij}\) :

Components of the extra tensor

\(z\) :

Upper wall of flow channel

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Acknowledgements

The authors sincerely appreciate funding from Researchers Supporting Project number (RSP2024R58), King Saud University, Riyadh, Saudi Arabia.

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Authors and Affiliations

  1. Department of Mathematics, Northern University, Wattar-Wallai Road, Nowshera, KPK, 24110, Pakistan

    Khurram Javid

  2. College of Engineering, King Saud University, Riyadh, Saudi Arabia

    Salahuddin Khan

  3. Pakistan Tokamak Plasma Research Institute (PTPRI), P. O. Box 3329, Islamabad, Pakistan

    Shahab Ud-Din Khan

  4. Ryan Institute’s Centre for Climate and Air Pollution Studies, Physics, School of Natural Sciences, University of Galway, Galway, Ireland

    Aqeel Afzal

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  1. Khurram Javid
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Javid, K., Khan, S., Khan, S.UD. et al. Slip and chemical reaction effects on the peristaltic rheology of a viscoplastic liquid in different wave frames: application of a high-permeability medium. Eur. Phys. J. Plus 139, 279 (2024). https://doi.org/10.1140/epjp/s13360-024-05015-3

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